Triangular ratio metric in the unit disk

Abstract

The triangular ratio metric is studied in a domain G R n , n ≥ 2. Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

Intrinsic metrics in ring domains

“…Acknowledgements This research continues my earlier work in [13][14][15][16][17]. Three last mentioned of these articles have been co-written with Professor Matti Vuorinen, to whom I am indebted for all guidance and support.…”

“…Any distinct points x, y ∈ R(r , 1) can be rotated around their midpoint in the following way so that the value of the triangular ratio metric for the rotated points can be found with either Proposition 3.4 or Lemma 3.6. Definition 3.9 [14,Def. 4.1, p. 10] Euclidean midpoint rotation.…”

“…Thus, in order to study the intrinsic geometry of the ring R(r , 1), we use here a few known generalizations of the B Oona Rainio ormrai@utu.fi 1 Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland hyperbolic metric: the j * -metric found first in [7] by modifying the distance ratio metric introduced in 1979 by Gehring and Palka [5], the triangular ratio metric introduced by Hästö in 2002 [9] and recently studied in [1,[14][15][16][17], and the Möbius metric originally introduced in [20, pp. 115-116] and extensively studied by P. Seittenranta in his PhD thesis [18].…”

Intrinsic Quasi-Metrics

“…Acknowledgements This research continues my work with Professor Matti Vuorinen in [14][15][16]. I am indebted to him for all guidance and other support.…”

Introducing a New Intrinsic Metric

“…More potential properties for a hyperbolic type metric are presented in [7, p. 191-192]. Examples of a hyperbolic type metric include the triangular ratio metric studied in [12][13][14], the Barrlund metric [4] and the Cassinian metric [10], whereas this work focuses on the following new intrinsic metric.…”